Math

A rectangular beam is to be cut from the log into a circular cross section. If the strength of the beam is proportional to the width and the square of the depth, find the dimensions that will give the strongest beam

  1. 👍 0
  2. 👎 0
  3. 👁 40
  1. I assume we can center the beam in the circle. So, let
    2x = width
    2y = depth
    That means that x^2+y^2 = r^2, the radius of the log
    If we let the strength be z, then we know that
    z = kxy^2 = kx(r^2-x^2) = kr^2x - kx^3
    dz/dx = k(r^2 - 3x^2)
    Now just find x where dz/dx=0
    You can see that z will be a maximum.

Respond to this Question

First Name

Your Response

Similar Questions

  1. math

    The stiffness S of a rectangular beam is proportional to its width (w) times the cube of its depth/height (h). find the dimensions (i.e.: w and h) of stiffest beam that can be cut from a log which has a circular cross-section with

    asked by deel on August 9, 2012
  2. Calculus

    The strength of a beam with rectangular corss-section is directly proportional to the product of the width and the square of the depth (thickness from the top to bottom of the beam). Find the shape of the strongest beam that can

    asked by Candy on March 10, 2012
  3. Calculus

    The strength of a beam with rectangular corss-section is directly proportional to the product of the width and the square of the depth (thickness from the top to bottom of the beam). Find the shape of the strongest beam that can

    asked by Liz on May 5, 2012
  4. Calculus

    I need help with this question: The strength of a beam with a rectangular cross section varies directly as x and as the square of y. What are the dimensions of the strongest beam that can be sawed out of a round log with diameter

    asked by John on February 25, 2011
  5. Statics

    A rectangular beam, with cross-sectional width b, height h, Young's modulus Erect, has the same flexural rigidity as a circular beam with the same cross-sectional area. What is the Young's modulus, Ecirc, of the circular beam in

    asked by qwerty on October 23, 2014
  6. Math

    The following table represents the diameter of the cross section of a wire at continuous heights (feet) above the ground. Assume that each cross section is circular. Height(ft) 2 6 10 14 18 22 26 30 Diameter 2 2 2.0 1.8 1.6 1.5

    asked by Paige on December 4, 2011
  7. Math

    The following table represents the diameter of the cross section of a wire at continuous heights (feet) above the ground. Assume that each cross section is circular. Height(ft) 2 6 10 14 18 22 26 30 Diameter 2 2 2.0 1.8 1.6 1.5

    asked by Paige on December 4, 2011
  8. calculus

    The strength of a rectangular beam is proportional to width*depth^2. What are the dimensions of the strongest rectangular beam that can be cut out of a 12 inch diameter log?

    asked by Susan on December 4, 2010
  9. mechanics of materials

    A steel beam with a rectangular cross section is bent to form an arc of a circle of radius 6 m. Calculate the maximum stress in the beam given that the depth of the beam is 6 mm and the Young's modulusfor steel is 210 MPa

    asked by dmkp on January 28, 2013
  10. Calculus1

    The strength, S, of a rectangular wooden beam is proportional to its width times the square of its depth. Find the dimensions of the strongest beam that can be cut from a 12 inch diameter cylindrical log.

    asked by ACDub on April 3, 2014
  11. Cross-Sectional Area

    The cross-sectional area of a beam cut from a log with radius 1 foot is given by the function A(x) = 4x(sqrt{1 - x^2}), where x represents the length of half the base of the beam. Determine the cross-sectional area of the beam if

    asked by John on February 22, 2008

More Similar Questions